Volume 21, Issue 3

Let C be a curve with two smooth components and a single node, and let 𝒰 C ( w , r , χ ) be the moduli space of w -semistable classes of depth one sheaves on C having rank r on both components and Euler characteristic χ . In this paper, under suitable assumptions,we produce a projective bundle over the product of the moduli spaces of semistable vector bundles of rank r on each component and we show that it is birational to an irreducible component of 𝒰 C ( w , r , χ ). Then we prove the rationality of the closed subset containing vector bundles with given fixed determinant.

We prove an integral formula for continuous paths of rectangles inscribed in a piecewise smooth loop. We use this integral formula to prove the inequality M ( γ ) ≥ Δ ( γ )/2 – 1, where M ( γ ) denotes the total multiplicity of rectangle coincidences, i.e. pairs, triples, etc. of isometric rectangles inscribed in γ , and Δ ( γ ) denotes the number of stable diameters of γ , i.e. critical points of the distance function on γ .

Let 𝔽 be the finite field of order q 2 . It is sometimes attributed to Serre that any curve 𝔽-covered by the Hermitian curveHq+1:yq+1=xq+x ${{\mathcal{H}}_{q+1}}:{{y}^{q+1}}={{x }^{q}}+x$is also 𝔽-maximal. For prime numbers q we show that every 𝔽-maximal curve x $\mathcal{x}$of genus g ≥ 2 with | Aut(𝒳) | > 84( g − 1) is Galois-covered by Hq+1. ${{\mathcal{H}}_{q+1}}.$The hypothesis on | Aut(𝒳) | is sharp, since there exists an 𝔽-maximal curve x $\mathcal{x}$for q = 71 of genus g = 7 with | Aut(𝒳) | = 84(7 − 1) which is not Galois-covered by the Hermitian curve H72. ${{\mathcal{H}}_{72}}.$

A cage is the 1-skeleton of a convex polytope in ℝ 3 . A cage is said to hold a set if the set cannot be continuously moved to a distant location, remaining congruent to itself and disjoint from the cage. In how many positions can (compact 2-dimensional) unit discs be held by a tetrahedral cage? We completely answer this question for all tetrahedra.

The moduli space M¯0,n(ℙ1,1) ${{\bar{M}}_{0,n}}\left( {{\mathbb{P}}^{1}},1 \right)$of n -pointed stable maps is a Mori dream space whenever the moduli space M¯0,n+3 of (n+3) ${{\bar{M}}_{0,n+3}}\; \text{of} \;(n+3)$pointed rational curves is, and M¯0,n(ℙ1,1) ${{\bar{M}}_{0,n}}\left( {{\mathbb{P}}^{1}},1 \right)$is a log Fano variety for n ≤ 5.

If G is a group acting geometrically on a CAT(0) cube complex X and if g ∈ G has infinite order, we show that exactly one of the following situations occurs: (i) g defines a rank-one isometry of X ; (ii) the stable centraliser SC G ( g ) = { h ∈ G ∣ ∃ n ≥ 1, [ h , g n ] = 1} of g is not virtually cyclic; (iii) Fix Y ( g n ) is finite for every n ≥ 1 and the sequence (Fix Y ( g n )) takes infinitely many values, where Y is a cubical component of the Roller boundary of X which contains an endpoint of an axis of g . We also show that (iii) cannot occur in several cases, providing a purely algebraic characterisation of rank-one isometries.

The hybrid model is the Landau–Ginzburg-type theory that is expected, via the Landau–Ginzburg/ Calabi–Yau correspondence, to match the Gromov–Witten theory of a complete intersection in weighted projective space. We prove a wall-crossing formula exhibiting the dependence of the genus-zero hybrid model on its stability parameter, generalizing the work of [21] for quantum singularity theory and paralleling the work of Ciocan-Fontanine–Kim [7] for quasimaps. This completes the proof of the genus-zero Landau– Ginzburg/Calabi–Yau correspondence for complete intersections of hypersurfaces of the same degree, as well as the proof of the all-genus hybrid wall-crossing [11].

Let F be the finite field of order q 2 . In this paper we continue the study in [24], [23], [22] of F -maximal curves defined by equations of type yn=xℓ (xm+1). ${{y}^{n}}={{x}^{\ell }}\left( {{x}^{m}}+1 \right).$New results are obtained via certain subcovers of the nonsingular model of vN=ut2−u ${{v}^{N}}={{u}^{{{t}^{2}}}}-u$where q = t α , α ≥ 3 is odd and N = ( t α + 1)/( t + 1). We observe that the case α = 3 is closely related to the Giulietti–Korchmáros curve.

Following Santaló’s approach, we prove several characterizations of a disc among bodies of constant width, constant projection lengths, or constant section lengths on given families of geodesics.

A Kakeya set 𝓚 in an affine plane of order q is the point set covered by a set 𝓛 of q + 1 pairwise non-parallel lines. By Dover and Mellinger [6], Kakeya sets with size at least q 2 – 3 q + 9 contain a large knot, i.e. a point of 𝓚 lying on many lines of 𝓛. We improve on this result by showing that Kakeya set of size at least ≈ q 2 – q q $\begin{array}{} \displaystyle \sqrt{q} \end{array}$ + 32 $\begin{array}{} \displaystyle \frac{3}{2} \end{array}$ q contain a large knot, and we obtain a sharp result for planes containing a Baer subplane.

We classify 3-dimensional compact Riemannian manifolds ( M 3 , g ) that admit a non-constant solution to the equation − Δfg +Hess f − f Ric = μ Ric + λg for some special constants ( μ , λ ), under the assumption that the manifold has cyclic parallel Ricci tensor. Namely, the structures that we study here are: positive static triples, critical metrics of the volume functional, and critical metrics of the total scalar curvature functional. We also classify n -dimensional critical metrics of the volume functional with non-positive scalar curvature and satisfying the cyclic parallel Ricci tensor condition.

In this article, we compute the motivic Igusa zeta function of a space monomial curve that appears as the special fiber of an equisingular family whose generic fiber is a complex plane branch. To this end, we determine the irreducible components of the jet schemes of such a space monomial curve. This approach does not only yield a closed formula for the motivic zeta function, but also allows to determine its poles. We show that, while the family of the jet schemes of the fibers is not flat, the number of poles of the motivic zeta function associated with the space monomial curve is equal to the number of poles of the motivic zeta function associated with a generic curve in the family.

We aim to classify the real hypersurfaces M in a Kaehler complex space form M n ( c ) satisfying the two conditions φl=lφ, $\varphi l=l\varphi ,$where l=R(⋅,ξ)ξ and φ $l=R(\cdot ,\xi )\xi \text{ and }\varphi $is the almost contact metric structure of M , and (∇ξl)X= $\left( {{\nabla }_{\xi }}l \right)X=$ ω ( X ) ξ , where where ω ( X ) is a 1-form and X is a vector field on M . These two conditions imply that M is a Hopf hypersurface and ω = 0.